Bottom Line:
A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible.We validate analytical results on four simulated network classes and empirical data sets of various sizes.Our treatment allows us to find support for Dunbar's hypothesis in detecting an upper threshold for the number of active social contacts that individuals maintain over the course of one week.

Affiliation: Department of Mathematics and Statistics, Vermont Complex Systems Center, The Computational Story Lab, and the Vermont Advanced Computing Core, University of Vermont, Burlington, Vermont, United States of America.

ABSTRACTComplex networks underlie an enormous variety of social, biological, physical, and virtual systems. A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible. Previous work addressing the impacts of partial network data is surprisingly limited, focuses primarily on missing nodes, and suggests that network statistics derived from subsampled data are not suitable estimators for the same network statistics describing the overall network topology. We generate scaling methods to predict true network statistics, including the degree distribution, from only partial knowledge of nodes, links, or weights. Our methods are transparent and do not assume a known generating process for the network, thus enabling prediction of network statistics for a wide variety of applications. We validate analytical results on four simulated network classes and empirical data sets of various sizes. We perform subsampling experiments by varying proportions of sampled data and demonstrate that our scaling methods can provide very good estimates of true network statistics while acknowledging limits. Lastly, we apply our techniques to a set of rich and evolving large-scale social networks, Twitter reply networks. Based on 100 million tweets, we use our scaling techniques to propose a statistical characterization of the Twitter Interactome from September 2008 to November 2008. Our treatment allows us to find support for Dunbar's hypothesis in detecting an upper threshold for the number of active social contacts that individuals maintain over the course of one week.

pone-0108471-g008: In, Out-degree vs. Average edge weight for Twitter reply networks.(Top, left) The average in-coming edge weight for each node of degree k is depicted in a logarithmically binned heatmap. (Top, right) The same as (a), except for out-going edges. (c.) The average weight per edge for in-coming edges as a function of kin shows a gradual increase to kin≈102 with a peak of approximately 2.2 interactions per edge. (d.) The average weight per edge for out-going edges as a function of kout shows a gradual increase to kout≈102 with a peak of between 2.5 and 3 interactions per edge.

Mentions:
Figure 7 depicts a log-log plot of the predicted node strength distribution. This plot reveals that there are fewer nodes in the high strength region than would be expected in a scale-free distribution. Figure 8 reveals that low degree nodes dominate the dataset and that many of these low degree nodes often have low average edge weight (). We find a peak in the average weight per edge as a function of degree around . This peak is more pronounced for out-going edges. Beyond this value, a limiting factor may prevent increases in the weight per edge, a result also noted by Gonçalves et al. [48].

pone-0108471-g008: In, Out-degree vs. Average edge weight for Twitter reply networks.(Top, left) The average in-coming edge weight for each node of degree k is depicted in a logarithmically binned heatmap. (Top, right) The same as (a), except for out-going edges. (c.) The average weight per edge for in-coming edges as a function of kin shows a gradual increase to kin≈102 with a peak of approximately 2.2 interactions per edge. (d.) The average weight per edge for out-going edges as a function of kout shows a gradual increase to kout≈102 with a peak of between 2.5 and 3 interactions per edge.

Mentions:
Figure 7 depicts a log-log plot of the predicted node strength distribution. This plot reveals that there are fewer nodes in the high strength region than would be expected in a scale-free distribution. Figure 8 reveals that low degree nodes dominate the dataset and that many of these low degree nodes often have low average edge weight (). We find a peak in the average weight per edge as a function of degree around . This peak is more pronounced for out-going edges. Beyond this value, a limiting factor may prevent increases in the weight per edge, a result also noted by Gonçalves et al. [48].

Bottom Line:
A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible.We validate analytical results on four simulated network classes and empirical data sets of various sizes.Our treatment allows us to find support for Dunbar's hypothesis in detecting an upper threshold for the number of active social contacts that individuals maintain over the course of one week.

Affiliation:
Department of Mathematics and Statistics, Vermont Complex Systems Center, The Computational Story Lab, and the Vermont Advanced Computing Core, University of Vermont, Burlington, Vermont, United States of America.

ABSTRACTComplex networks underlie an enormous variety of social, biological, physical, and virtual systems. A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible. Previous work addressing the impacts of partial network data is surprisingly limited, focuses primarily on missing nodes, and suggests that network statistics derived from subsampled data are not suitable estimators for the same network statistics describing the overall network topology. We generate scaling methods to predict true network statistics, including the degree distribution, from only partial knowledge of nodes, links, or weights. Our methods are transparent and do not assume a known generating process for the network, thus enabling prediction of network statistics for a wide variety of applications. We validate analytical results on four simulated network classes and empirical data sets of various sizes. We perform subsampling experiments by varying proportions of sampled data and demonstrate that our scaling methods can provide very good estimates of true network statistics while acknowledging limits. Lastly, we apply our techniques to a set of rich and evolving large-scale social networks, Twitter reply networks. Based on 100 million tweets, we use our scaling techniques to propose a statistical characterization of the Twitter Interactome from September 2008 to November 2008. Our treatment allows us to find support for Dunbar's hypothesis in detecting an upper threshold for the number of active social contacts that individuals maintain over the course of one week.